Usage patterns
〈人〉に長男が生まれるsomeone's eldest son is born
His eldest son was born last year.
長男として育つto grow up as the eldest son (often with expectations)
He was raised strictly as the eldest son.
長男が家を継ぐthe eldest son inherits/succeeds the family (business, household)
In this family, it's expected that the eldest son will take over the family business.